Apr 29, 2013 by Stuart Mason Dambrot

Snapshots of the algorithm starting with a sphere and finding the cube from its spectrum alone. Courtesy and acknowledgement Achim Kempf. Copyright 2013 by The American Physical Society.

(Phys.org) —Mathematics is, in essence, an artificial language for precisely articulating theories about the physical world. Unlike natural language, however, translating different classes of mathematics can be difficult at best. Such is the case encountered in the attempt to unify general relativity and quantum theory, since they are expressed in differential geometry and functional analysis, respectively. That being said, spectral geometry – a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators – may resolve this long-standing quandary by allowing spacetime to be treated as simultaneously continuous and discrete, essentially relating the frequency-based ringing of the fabric of spacetime to its manifold-based shape. Recently, scientists at California Institute of Technology, Princeton University, University of Waterloo, and University of Queensland normalized and segmented spectral geometry into small, finite-dimensional steps. They then demonstrated their approach of calculating the shapes of two-dimensional objects from their vibrational spectra as being viable in two, and possibly more, dimensions.

Prof. Achim Kempf discussed the research he, David Aasen, Tejal Bhamre conducted. "Before the new results," Kempf tells Phys.org, "it was thought that spectral geometry is too nonlinear – and therefore simply too hard to use – for the purpose of unifying general relativity and quantum theory. In the new paper, however, we showed that spectral geometry can be tamed and made into a very useful practical method, namely by suitably cutting it into small linear, and therefore manageable, pieces."

Kempf notes that in special cases, spectral geometry has certain ambiguities: mathematically, such as special curved shapes in high dimensions that have the same spectrum – that is, they would sound the same of we could detect higher dimensions. "The worry has been that if there were too many such ambiguities also in our three-dimensional world, this could make spectral geometric impractical as a tool in physics," Kempf explains. "In the new paper we showed that, fortunately, the small linearized steps that we take are almost always ambiguity-free – and for two dimensional shapes in three dimensions, we didn't find ambiguities at all. Relatedly, it would be very interesting to extend spectral geometry from a description of space at each time to a unified description of both space and time. This still needs to be developed further."

That being said, Kempf points out that their idea – addressing spectral geometry's difficulty and ambiguities by regularizing and segmenting spectral geometry into finite-dimensional steps – works very well. "The computation time can be a little long," he notes, but we think that we will be able to significantly speed up the calculations. We'd like to be able to run them, for example, on a smartphone."

A single key insight enabled the researchers addressed these challenges in two ways. Essentially, as far as the mathematics is concerned, the problem was to find a method that would allow one to calculate the shape of an object from the sound that it makes when vibrating. "To this end, the key insight was that this spectral geometric problem, in spite of being highly nonlinear, can actually be tamed with our strategy, which has two components," Kempf explains. "First, make the nonlinear calculations manageable by cutting them into small doable steps." In practice, he notes that the computer does this by starting with some random shape, such as the shape of a sphere. Then, while it keeps comparing the sound of the sphere to the sound of the object that it needs to identify, the computer will change the shape of the sphere until it reaches the shape of the object that it had to identify.

"The second step is to regularize – that is, don't try to get all of the shape's details at once," Kempf says. "Instead, calculate the rough shape from just part of the sound spectrum." By then incrementally using more of the sound spectrum, this approach allows them to specify the shape with increasing accuracy.

"The beauty of our new spectral geometry is that it allows us to describe the shape of a vase, or eventually the shape of the fabric of spacetime, through so-called invariants – that is, by quantities that do not depend on any choice of coordinate system," Kempf adds. "This is important because if we're to develop a theory that unifies quantum theory and general relativity, key quantities fundamentally cannot depend on man-made choices, such as which coordinate system one wants to use."

Kempf then summarized the relation of their approach, which offers a gauge-independent identification of the metric's degrees of freedom in terms of invariants that should be ready to quantize, with several other mathematical attempts to unify general relativity and quantum theory.

Loop quantum gravity and string theory: "The new spectral geometric methods are deeply related to generalized Heisenberg uncertainty principles – and in fact, the new work grew out of studies of such principles, which have been shown to be related to loop quantum gravity as well as to string theory by myself and in collaboration with Martin Bojowald1."

Causal sets: "Perhaps, but it's not clear if there's a connection."

Garrett Lisi's E8 proposal: "Probably no connection."

Noncommutative geometry: "Alain Connes' program of noncommutative geometry shows that curved spaces can be described by a spectral triple, which includes the spectrum of the Dirac operator. It's not clear if the spectrum of the Dirac operator alone is sufficient to calculate the shape of a curved space. The new spectral geometric methods that we present here can be used to explore this interesting question further, and in fact we're working on this."

Supergravity: "Our new results apply to gravity and do not require supersymmetry. This is good because there's still no solid evidence that supersymmetry exists in nature."

Twistor models: "No connection known."

Moving forward, says Kempf, the scientists are working on generalizing the new methods to shapes that are curved in both space and time, since that will then be useful for addressing some of the key questions of cosmology – including the question of how it all began." More specifically, Kempf adds that while quantum fluctuations are today almost immeasurably small, it's thought that spacetime itself arose from a kind of quantum jump. "Our results bring us a step closer to being able to explicitly calculate the quantum ringing of spacetime, which could then tell us more about the origin of our universe."

In terms of other areas of research that might benefit from their study, Kempf points out that experimentalists still have a long way to go to measure quantum gravity effects directly. "However," he adds, "our new methods can also be used to program a computer to calculate the shape of objects from their sound. Moreover," he concludes, "we're planning to improve our algorithm to make it much faster. This could open up engineering applications, for example, by allowing machines to quickly identify shapes from a simple spectral fingerprint."

Related:
1Generalized uncertainty principles and localization of a particle in discrete space. Physical Review D 86, 085017 (2012), DOI:10.1103/PhysRevD.86.085017Spacetime could be simultaneously continuous and discrete, in the same way that information can be. New Journal of Physics 12 115001 (2010), doi:10.1088/1367-2630/12/11/115001

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It's an interesting approach. Sort of like they do in medical imaging (CT, MR, ..): You look at the projections from various angles and then try to backpropagate the information - refining iteratively to get at the correct 'shape' of the underlying reality.

Only that here you don't have a set of images but a set of formulas.

If the analogy with 3D imaging holds then it should give a rather good approximation. However, it would also point to a fundamental inability to get the absolutely correct 'shape' as no amount of projections will give you enough information to reconstruct a volume 100% correctly.That is: unless you also know that the individual units of the shape are quantized and what the quantizations are - in that case you could get a perfect picture. But preknowledge about that seems rather hard to come by with formulas.

All modern physics basec on abstract mathematics is wrong.Relativity theory is wrong and quantum mechanics.Spacetime doesn't exist.In Lorentz equations x and t are not space and time but wavelength and period. The Lorentz equations give the Doppeler shift for Transversal waves.

Sounds interesting. As usual, PhysOrg assumes we all have access to journals. Not True !http://arxiv.org/...12.5297. Articles published in Nature or Science have no preprints.A very odd statement is made by Kempf:"Supergravity: Our new results apply to gravity and do not require supersymmetry. This is good because there's still no solid evidence that supersymmetry exists in nature". Very true. This is one of its most compelling features. SUSY is D.O.A. in nature, yet theorists desperately continue to kluge new versions of it, trying to explain null experimental searches for sparticles at LHC.Yet they partially attribute connexions of their result to string theory, which depends Critically on the existence of SUSY, and in fact collapses without it, as does M-theory.So, something's got to give.

"Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality." Nikola Tesla

Tah-da, no relation to reality!

Nicolas Tesla also believed that atoms were not composed of subatomic particles.

Just because you don't understand the theory involved doesn't mean that it has no relationship to reality. On the contrary, both general relativity and quantum theory have been verified to high degree, and came, ironically, from our mathematical theory NOT agreeing with reality (for example, the ultraviolet paradox).

If you did some research on this, you would see that the math in this article is easily expressed in terms of reality. Essentially, it deals with extracting the geometric shape of an object from the way it resonates. The wave spectra of snare drums and trumpets differ greatly due to their shapes. In this case, that object is space-time.

"Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality." Nikola Tesla

Tah-da, no relation to reality!

This is indeed often the case - but it's nothing new, as Ptolemaic cosmology demonstrates. Moreover, in the scientific framework theory precedes hypothesis-based testing, which often has to wait until technology that can address that level of scale is available. One of the most salient examples of this is, of course, Einstein, who started with a thought experiment and then learned the mathematics needed to precisely formulate his theories (not just relativity, but also the foundations for quantum mechanics - the two theories these researchers are trying to unify - even though Einstein could not accept his own results implying entanglement). Nonetheless, relativity has yet to be disproven - which is the purpose of the scientific method.

Nonetheless, relativity has yet to be disproven - which is the purpose of the scientific method.

"It is important to understand that while a theory may permit observations, those observations do not necessarily verify the theory." Anon

This idea relativity has yet to be "disproven" is completely false. Many men have come forth showing numerous aspects of GR to have been falsified. Most in the astrophysical refuse to acknowledge such facts since they tend to not understand the theory itself. "Since the mathematicians have invaded the theory of relativity, I do not understand it myself any more." EinsteinGR has been a multi-billion dollar thought experiment that has stagnated true scientific progress for a century now, it will one day fade as epicycles did. It's very likely our descendants will look to the current era of the "dark ages" with incredible contempt, similar to the flat and hollow earthers of today.

This idea relativity has yet to be "disproven" is completely false. Many men have come forth showing numerous aspects of GR to have been falsified.

Who?

GR has been a multi-billion dollar thought experiment that has stagnated true scientific progress for a century now

How has it stagnated true scientific progress? We've discovered all kinds of awesome things using GR, like black holes, that the universe is accelerating, that galaxies seem to contain more mass than they should, and even some exoplanets!

it will one day fade as epicycles did. It's very likely our descendants will look to the current era of the "dark ages" with incredible contempt, similar to the flat and hollow earthers of today.

In your humble opinion, no doubt. Do you have an alternative explanation that can be experimentally tested?

We've discovered all kinds of awesome things using GR, like black holes, that the universe is accelerating, that galaxies seem to contain more mass than they should, and even some exoplanets!

All of which remain hypothetical "discoveries" based solely on interpretations of the data made by astrophysicists whose only background is GR due to the required specialization of the current scientific paradigm.

Nicolas Tesla also believed that atoms were not composed of subatomic particles

I seriously doubt it: In his early 1890s lectures at Columbia University, the Chicago World's Fair and at Royal Societies in Paris and London, building on the ideas of Isaac Newton and Lord Kelvin, Tesla demonstrated and discussed the structure of atoms as being similar to solar systems and wave-like and particle-like aspects to what later became known as the photon.

All of which remain hypothetical "discoveries" based solely on interpretations of the data made by astrophysicists

No, they are discoveries whether you like them or not, and unless you can fly there, or directly image them, they're the best we have at the moment. You still haven't answered my challenge. Do you have anything better that can be experimentally tested, or do you just want to deride a theory to feel superior?

whose only background is GR due to the required specialization of the current scientific paradigm.

Kempf then summarized the relation of their approach, which offers a gauge-independent identification of the metric's degrees of freedom in terms of invariants that should be ready to quantize, with several other mathematical attempts to unify general relativity and quantum theory.Loop quantum gravity and string theory, Supergravity, supersymmetry, etc. …….

It seems that all the proposed ideas are too complicate and difficult to understand, maybe this simple physical view could guide us how to unify GR and QM…http://www.vacuum...=9〈=en

All of which remain hypothetical "discoveries" based solely on interpretations of the data made by astrophysicists whose only background is GR due to the required specialization of the current scientific paradigm.

You display a fundamental misunderstanding of how scientists work. We take the data and then apply it to theory, not the other way around. If a theory does not accurately describe the data, then we adjust the theory to fit the data.

All of which remain hypothetical "discoveries" based solely on interpretations of the data made by astrophysicists whose only background is GR due to the required specialization of the current scientific paradigm.

You display a fundamental misunderstanding of how scientists work. We take the data and then apply it to theory, not the other way around. If a theory does not accurately describe the data, then we adjust the theory to fit the data.

In this case, GR is supported by the data.

If GR is all they have been "educated" with, how in the world would they propose an alternative? Maybe that's why so many astrophysical articles relate to the surprise and unexpectedness of so many observations. When a theory is so completely "adjustable" as GR is, it's no wonder the theory is so "successful"!

Like with asymptotic safety, this seems to be the directly wrong way to go about resolving this issue. "If we fiddle with the math in the right way, we'll be able to get two systems that we can only describe in two physically different mannerisms to calculate with each other, that solves everything!" But the universe, I'm sure, doesn't care how we describe things. It's not simultaneously discretized and continuous at the same time, not matter how much we want it to be. Physical reality is one or the other, and to get the best description of it possible we're going to have to figure out which one it is.

Good luck finding an N dimensional blackboard to draw the results;-) But I do like the approach. As we cannot measure any higher dimensions directly: Guess and check then re-guess and re-check, just might yield an unexpected avenue for the next 'Theory Of Everything' step .

When a theory is so completely "adjustable" as GR is, it's no wonder the theory is so "successful"!

You are confused. Nobody is adjusting GR (well, except for maybe determining the exact value of the cosmological constant). Rather, assuming that GR is true, people have detected things such as black holes, which are consistent with what GR predicts. Now, dark matter is a different story. GR cannot predict its existence or nonexistence. If you want to call Cosmology adjustable, that's another argument to be had.Here's how it is: GR is the best we have right now concerning gravity. Contrary to what you believe, it has been tested against competing theories many times and has always come out on top. Don't like it? Propose a new theory that can be tested, or carry out an experiment that disagrees with its predictions.

All of which remain hypothetical "discoveries" based solely on interpretations of the data made by astrophysicists whose only background is GR due to the required specialization of the current scientific paradigm.

You display a fundamental misunderstanding of how scientists work. We take the data and then apply it to theory, not the other way around. If a theory does not accurately describe the data, then we adjust the theory to fit the data.

In this case, GR is supported by the data.

I don't mean to interrupt,,, but I have to know. Are ya THE Zephyr? Or some other zephyr? Just curious. Carry on.

...Here's how it is: GR is the best we have right now concerning gravity. Contrary to what you believe, it has been tested against competing theories many times and has always come out on top. Don't like it? Propose a new theory that can be tested, or carry out an experiment that disagrees with its predictions.

I completely agree . There's an unsettling regression to, or resurgence of, pre-scientific thought that is increasing along with, or perhaps in reaction to, the scientific method being the best way yet for us to understand the universe as it is, not as we dream it to be.

Let me summarize: an idea is not a theory, but can give rise to one, ahd facts are not determined by inductive reasoning alone. A hypothesis is a specific experimental design for testing a theory's validity, the aim being to determine where it falls short - and if it does in small ways, it may be correctable, but if enough falsities occur it can be considered incorrect (i.e., disproved).

As a layman I'm pretty new to SG but I'm interested in this idea of trying to find similarities between one branch of science and another with the aim of unification. I have read about such ideas on higher dimensions(C.McMullen, R. Rucker) but this is the first bit of 'research' I've seen. As far as the universe 'singing' that's an old phrase but I get the feeling that the meaning here is not the same. I wonder what kind of 'music' we might 'hear' in say the 4th dimension...not 'punk rock' I hope.Would it be possible to correlate, say, the harmonics with the equations for finding the number of lines (squares, cubes etc.) in the higher geometrical figures?

All of which remain hypothetical "discoveries" based solely on interpretations of the data made by astrophysicists whose only background is GR due to the required specialization of the current scientific paradigm.

You display a fundamental misunderstanding of how scientists work. We take the data and then apply it to theory, not the other way around. If a theory does not accurately describe the data, then we adjust the theory to fit the data.

In this case, GR is supported by the data.

Hate to correct your theory, but - Scientists had nothing to test until Einstein provided the Theory of General Relativity. I will allow you that he was inspired by experiments of others to create that theory. However, that collation gave way to a mathematical model which then set about testing it.And your 2 sentences above contradict each other.Perhaps we should stop calling it a theory BEFORE experiments are done and just call it an idea supported by mathematical framework.

The principles of Special then General Relativity flowed mathematically from the earlier 'principle of relativity': That physical laws are independent of your frame of reference. Once the speed of light had been proved to be independent of your frame of reference (Moving or otherwise) the mathematical derived principles Einstein made were inevitable. Applications of these relativity principles are now routine. GPS timing corrections and gravitational lensing and E=mc^2 flow inevitably and dependably from these principles: Regardless of whether Tesla or any of the dissenting contributors here like it or not.

I think some of the disagreement here comes from nomeclature. In a scientific application some terms have different definitions than in common language usage, such as:

Law: Something that has been thoroughly tested and can be reproduced in a lab or other controlled setting. (Entropy, conservation of momentum.) Laws ALWAYS apply in any setting there are no loopholes.

Theory: Something that has been thoroughly tested but which cannot be reproduced in a lab or controlled setting. (Relativity, evolution, big bang.) Theories are nearly as absolute as laws however they don't always hold true in every circumstance or break down slightly when measured with enough presision. A theory could be viewed as a law that may have a loophole or two.

Hypothesis: This is an idea that will be tested by experiment and either be confirmed, modified, or discarded depending on the results of the experiments. This is much closer to the common usage of the word theory.

: Something that has been thoroughly tested but which cannot be reproduced in a lab or controlled setting. (Relativity, evolution, ...

Just a niggle: Evolution and relativity can be (and have been) tested in the lab.Mutation and selection is rather easy to test. And for relativity one just has to look at a piece of gold (without the relativistic effect on the electrons it would appear silver like most other metals. If one includes GPS as a 'laboratory setting' then certainly relativistic effects are apparent - without taken them into account it would be much less accurate.)

In dense aether model the mutual connection of quantum mechanics and general relativity can be understood with water surface analogy of space-time. Try to imagine, you're representing a piece of foam floating at the water surface. After then you have essentially two basic options, how to observe it with its own waves in most deterministic ways possible. The relativistic way considers, that the spreading of ripples is independent to (motion/reference frame) of underwater. Only subtle deforms/density fluctuations are allowed and they will manifest itself as a lensing focusing the surface waves.

The quantum mechanic way is relevant to small distance scales, where the water is behaving like dense particle system full of chaotic density fluctuations blurred with Brownian motion, which are behaving like the foam, the density of which is proportional to energy density in each place and time interval. This is the quantitative basis of quantum mechanics, the Schroedinger equation in particular.

@grondiluYou will learn more looking up Special Relativity on Wikipedia than in these posts.http://en.wikiped...lativityThe comments often devolve into a soapbox for peoples' pet theories. I sometimes think I've accidentally walked in on a bunch of Scientologists and UFO conspiracy theorists.

One of the easy to follow consequences is, the hypothetical worlds observed at water surface with its own waves are geometrically self-similar. The surface ripples spread in regular circles first, but they're scattering at distance in mixture of solitons, which do propagate like the particles. The geometrically similar process happens at the small scale, where the surface ripples are getting scattered with Brownian noise. And they undergo the packing geometry typical for particles described with Lie gauge groups. The physicists already recognized in during study of hypedimensional models of relativity and quantum mechanics, that they're geometrically similar, just inverted mutually. They're calling it AdS/CFT correspondence.

One testable consequence of this correspondence is the geometric similarity of density fluctuations of vacuum at largest scale (which are called the dark matter foam and they're observable with fluctuations at the CMBR noise) and the density fluctuations of vacuum at the smallest scale (which are called the quantum foam and they're observable with fluctuation at the Higgs field as a Higgs bosons). The power spectrum of CMBR spectrum is geometrically selfsimilar to energy spectrum of Higgs boson. They're IMO manifestation of the same dodecahedral structure of foam, which is typical for large systems of packed particles. Because the power spectra of these geometries is self-similar at both largest, both smallest scales, it represents the way, in which the models of general relativity could be reconciled with model of quantum mechanics. This is IMO the rational physical basis of the above study.